a kastler kalau walze type theorem and the spectral action for perturbations of dirac operators on manifolds with boundary

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 619120, 13 pages http://dx.doi.org/10.1155/2014/619120 Research Article A Kastler-Kalau-Walze Type Theorem and the Spectral Action for Perturbations of Dirac Operators on Manifolds with Boundary Yong Wang School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China Correspondence should be addressed to Yong Wang; wangy581@nenu.edu.cn Received 23 October 2013; Accepted 13 January 2014; Published 17 March 2014 Academic Editor: Jaume Gin´e Copyright © 2014 Yong Wang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds Introduction The noncommutative residue found in [1, 2] plays a prominent role in noncommutative geometry In [3], Connes used the noncommutative residue to derive a conformal 4dimensional Polyakov action analogy In [4], Connes proved that the noncommutative residue on a compact manifold 𝑀 coincided with Dixmier’s trace on pseudodifferential operators of order − dim 𝑀 Several years ago, Connes made a challenging observation that the noncommutative residue of the square of the inverse of the Dirac operator was proportional to the Einstein-Hilbert action, which is called the Kastler-Kalau-Walze theorem now In [5], Kastler gave a brute-force proof of this theorem In [6], Kalau and Walze proved this theorem by the normal coordinates way simultaneously In [7], Ackermann gave a note on a new proof of this theorem by the heat kernel expansion way The KastlerKalau-Walze theorem had been generalized to some cases, for example, Dirac operators with torsion [8], CR manifolds [9], and R𝑛 [10] (see also [11, 12]) On the other hand, Fedosov et al defined a noncommutative residue on Boutet de Monvel’s algebra and proved that it was the unique continuous trace in [13] In [14], Schrohe gave the relation between the Dixmier trace and the noncommutative residue for manifolds with boundary In [15, 16], we gave an operator-theoretic explanation of the gravitational action for manifolds with boundary and proved a KastlerKalau-Walze type theorem for Dirac operators and signature operators on manifolds with boundary Perturbations of Dirac operators were investigated by several authors In [17], Sitarz and Zajac investigated the spectral action for scalar perturbations of Dirac operators In [18, p 305], Iochum and Levy computed the heat kernel coefficients for Dirac operators with one-form perturbations In [19], Hanisch et al derived a formula for the gravitational part of the spectral action for Dirac operators on 4-dimensional spin manifolds with totally antisymmetric torsion and this is a perturbation with three forms of Dirac operators On the other hand, in [20], Connes and Moscovici considered the conformal perturbations of Dirac operators Investigating the perturbations of Dirac operators has some significance (see [18, 19, 21]) Motivated by [17–19], we study the Dirac operators with general form perturbations We prove a Kastler-Kalau-Walze type theorem for general form perturbations and the conformal perturbations of Dirac operators for compact manifolds with or without boundary We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact manifolds and give detailed computations of spectral action for scalar perturbations of Dirac operators in [17] 2 Abstract and Applied Analysis This paper is organized as follows In Section 2, we prove the Lichnerowicz formula for perturbations of Dirac operators and prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on 4-dimensional compact manifolds with or without boundary In Section 3, we prove a Kastler-Kalau-Walze type theorem for conformal perturbations of Dirac operators on compact manifolds with or without boundary In Section 4, we compute the spectral action for Dirac operators with scalar and two-form perturbations on 4-dimensional compact manifolds 2.1 A Kastler-Kalau-Walze Type Theorem for Perturbations of Dirac Operators on Manifolds without Boundary Let 𝑀 be a smooth compact Riemannian 𝑛-dimensional manifold without boundary and let 𝑉 be a vector bundle on 𝑀 Recall that a differential operator 𝑃 is of Laplace type if it has locally the form 𝑖 𝑃 = − (𝑔 𝜕𝑖 𝜕𝑗 + 𝐴 𝜕𝑖 + 𝐵) , 𝑃 = − [𝑔 (∇𝜕𝑖 ∇𝜕𝑗 − ∇∇𝜕𝐿 𝜕𝑗 ) + 𝐸] , 𝑖 (3) 𝐸 = 𝐵 − 𝑔𝑖𝑗 (𝜕𝑖 (𝜔𝑗 ) + 𝜔𝑖 𝜔𝑗 − 𝜔𝑘 Γ𝑖𝑗𝑘 ) , where Γ𝑖𝑗𝑘 are the Christoffel coefficients of ∇𝐿 Now, we let 𝑀 be an 𝑛-dimensional oriented spin manifold with Riemannian metric 𝑔 We recall that the Dirac operator 𝐷 is locally given as follows in terms of orthonormal frames 𝑒𝑖 , ≤ 𝑖 ≤ 𝑛, and natural frames 𝜕𝑖 of 𝑇𝑀: 𝐷 = ∑𝑔𝑖𝑗 𝑐 (𝜕𝑖 ) ∇𝜕𝑆𝑗 = ∑𝑐 (𝑒𝑖 ) ∇𝑒𝑆𝑖 , 𝑖,𝑗 𝑖 (4) where 𝑐(𝑒𝑖 ) denotes the Clifford action which satisfies the relation 𝑐 (𝑒𝑖 ) 𝑐 (𝑒𝑗 ) + 𝑐 (𝑒𝑗 ) 𝑐 (𝑒𝑖 ) = 𝑗 −2𝛿𝑖 , ∇𝜕𝑆𝑖 = 𝜕𝑖 + 𝜎𝑖 , 𝜎𝑖 = ∑ ⟨∇𝜕𝐿𝑖 𝑒𝑗 , 𝑒𝑘 ⟩ 𝑐 (𝑒𝑗 ) 𝑐 (𝑒𝑘 ) 𝑗,𝑘 Γ𝑘 = 𝑔𝑖𝑗 Γ𝑖𝑗𝑘 (6) By (6a) in [5], we have 𝐷2 = − 𝑔𝑖𝑗 𝜕𝑖 𝜕𝑗 − 2𝜎𝑗 𝜕𝑗 + Γ𝑘 𝜕𝑘 − 𝑔𝑖𝑗 [𝜕𝑖 (𝜎𝑗 ) + 𝜎𝑖 𝜎𝑗 − Γ𝑖𝑗𝑘 𝜎𝑘 ] + 𝑠, (𝐷 + Ψ)2 = 𝐷2 + 𝐷Ψ + Ψ𝐷 + Ψ2 , (7) (5) (8) 𝐷Ψ + Ψ𝐷 = ∑𝑔𝑖𝑗 (𝑐 (𝜕𝑖 ) Ψ + Ψ𝑐 (𝜕𝑖 )) 𝜕𝑗 𝑖𝑗 + ∑𝑔𝑖𝑗 (𝑐 (𝜕𝑖 ) 𝜕𝑗 (Ψ) + 𝑐 (𝜕𝑖 ) 𝜎𝑗 Ψ (9) 𝑖𝑗 + Ψ𝑐 (𝜕𝑖 ) 𝜎𝑗 ) By (7)–(9), we have 𝐷Ψ = − 𝑔𝑖𝑗 𝜕𝑖 𝜕𝑗 + (−2𝜎𝑗 + Γ𝑗 + 𝑐 (𝜕𝑗 ) Ψ + Ψ𝑐 (𝜕𝑗 )) 𝜕𝑗 + 𝑔𝑖𝑗 [−𝜕𝑖 (𝜎𝑗 ) − 𝜎𝑖 𝜎𝑗 + Γ𝑖𝑗𝑘 𝜎𝑘 + 𝑐 (𝜕𝑖 ) 𝜕𝑗 (Ψ) (10) + 𝑐 (𝜕𝑖 ) 𝜎𝑗 Ψ + Ψ𝑐 (𝜕𝑖 ) 𝜎𝑗 ] + 𝑠 + Ψ2 (2) where ∇𝐿 denotes the Levi-Civita connection on 𝑀 Moreover (with local frames of 𝑇∗ 𝑀 and 𝑉), ∇𝜕𝑖 = 𝜕𝑖 + 𝜔𝑖 and 𝐸 are related to 𝑔𝑖𝑗 , 𝐴𝑖 , and 𝐵 through 𝑗 𝜔𝑖 = 𝑔𝑖𝑗 (𝐴𝑗 + 𝑔𝑘𝑙 Γ𝑘𝑙 𝐼𝑑) , 𝜎𝑖 = 𝑔𝑖𝑗 𝜎𝑗 , (1) where 𝜕𝑖 is a natural local frame on 𝑇𝑀, 𝑔𝑖,𝑗 = 𝑔(𝜕𝑖 , 𝜕𝑗 ) and (𝑔𝑖𝑗 )1≤𝑖,𝑗≤𝑛 is the inverse matrix associated with the metric matrix (𝑔𝑖,𝑗 )1≤𝑖,𝑗≤𝑛 on 𝑀, and 𝐴𝑖 and 𝐵 are smooth sections of End(𝑉) on 𝑀 (endomorphism) If 𝑃 is a Laplace type operator of the form (1), then (see [22]) there is a unique connection ∇ on 𝑉 and a unique endomorphism 𝐸 such that 𝑖𝑗 𝜕𝑗 = 𝑔𝑖𝑗 𝜕𝑖 , where 𝑠 is the scalar curvature Let Ψ be a smooth differential form on 𝑀 and we also denote the associated Clifford action := (𝐷 + Ψ)2 We note that by Ψ We will compute 𝐷Ψ A Kastler-Kalau-Walze Type Theorem for Perturbations of Dirac Operators 𝑖𝑗 Let By (10) and (3), we have 𝜔𝑖 = 𝜎𝑖 − [𝑐 (𝜕𝑖 ) Ψ + Ψ𝑐 (𝜕𝑖 )] , 𝐸 = − 𝑐 (𝜕𝑖 ) 𝜕𝑖 (Ψ) − 𝑐 (𝜕𝑖 ) 𝜎𝑖 Ψ − Ψ𝑐 (𝜕𝑖 ) 𝜎𝑖 1 − 𝑠 − Ψ2 + 𝜕𝑗 [𝑐 (𝜕𝑗 ) Ψ + Ψ𝑐 (𝜕𝑗 )] 1 − Γ𝑘 [𝑐 (𝜕𝑘 ) Ψ + Ψ𝑐 (𝜕𝑘 )] + 𝜎𝑗 [𝑐 (𝜕𝑗 ) Ψ + Ψ𝑐 (𝜕𝑗 )] 2 + [𝑐 (𝜕𝑗 ) Ψ + Ψ𝑐 (𝜕𝑗 )] 𝜎𝑗 − 𝑔𝑖𝑗 [𝑐 (𝜕𝑖 ) Ψ + Ψ𝑐 (𝜕𝑖 )] [𝑐 (𝜕𝑗 ) Ψ + Ψ𝑐 (𝜕𝑗 )] (11) For a smooth vector field 𝑋 on 𝑀, let 𝑐(𝑋) denote the Clifford action So, ∇𝑋 = ∇𝑋𝑆 − [𝑐 (𝑋) Ψ + Ψ𝑐 (𝑋)] (12) Since 𝐸 is globally defined on 𝑀, so we can perform computations of 𝐸 in normal coordinates Taking normal Abstract and Applied Analysis coordinates about 𝑥0 , then, 𝜎𝑖 (𝑥0 ) = 0, 𝜕𝑗 [𝑐(𝜕𝑗 )](𝑥0 ) = 𝑘 𝑖𝑗 0, Γ (𝑥0 ) = 0, 𝑔 (𝑥0 ) = 𝑗 𝛿𝑖 , so that ∇𝑌 = ∇𝑌𝑆 + √−1𝑔(𝑋, 𝑌), where 𝑌 is a smooth vector field on 𝑀 By 𝑒𝑗 (𝑐(𝑒𝑖 )) = and 𝑑𝑒𝑙 (𝑥0 ) = (see [15]), we have √−1 𝐸 (𝑥0 ) = − 𝑠 − |𝑋|2 + 1 𝐸 (𝑥0 ) = − 𝑠 − Ψ2 + [𝜕𝑗 (Ψ) 𝑐 (𝜕𝑗 ) − 𝑐 (𝜕𝑗 ) 𝜕𝑗 (Ψ)] × [𝑒𝑗 (𝑎𝑘 ) 𝑐 (𝑒𝑘 ) 𝑐 (𝑒𝑗 ) − 𝑐 (𝑒𝑗 ) 𝑐 (𝑒𝑘 ) 𝑒𝑗 (𝑎𝑘 )] − [𝑐 (𝜕𝑖 ) Ψ + Ψ𝑐 (𝜕𝑖 )] (𝑥0 ) + [𝑐 (𝑒𝑖 ) 𝑐 (𝑋) + 𝑐 (𝑋) 𝑐 (𝑒𝑖 )] 1 = − 𝑠 − Ψ2 + [𝑒𝑗 (Ψ) 𝑐 (𝑒𝑗 ) − 𝑐 (𝑒𝑗 ) 𝑒𝑗 (Ψ)] √−1 = − 𝑠+ 𝑒 (𝑎𝑘 ) [𝑐 (𝑒𝑘 ) 𝑐 (𝑒𝑗 ) − 𝑐 (𝑒𝑗 ) 𝑐 (𝑒𝑘 )] 𝑗 − [𝑐 (𝑒𝑖 ) Ψ + Ψ𝑐 (𝑒𝑖 )] (𝑥0 ) = − 𝑠 + √−1 ∑ 𝑒𝑗 (𝑎𝑘 ) 𝑐 (𝑒𝑘 ) 𝑐 (𝑒𝑗 ) (𝑥0 ) 𝑘 ≠ 𝑗 1 = − 𝑠 − Ψ2 + [∇𝑒𝑆𝑗 (Ψ) 𝑐 (𝑒𝑗 ) − 𝑐 (𝑒𝑗 ) ∇𝑒𝑆𝑗 (Ψ)] 2 − [𝑐 (𝑒𝑖 ) Ψ + Ψ𝑐 (𝑒𝑖 )] (𝑥0 ) Corollary For a one-form 𝜂 and the Clifford action 𝑐(𝜂), one has Proposition Let Ψ be a smooth differential form on 𝑀 and 𝐷Ψ := 𝐷 + Ψ; then (14) where ∇𝜕𝑖 is defined by (12) and 𝑋 = 𝜕𝑖 ∇𝑋 = − 𝑓𝑐 (𝑋) , (18) When Ψ is a two-form, we let Ψ = ∑𝑘